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\title{Deciphering the Heartbeat of Turbofan Engines: Exploring Smart Predictive Maintenance and Decision-Making}  % 标题

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% 此处填写摘要内容
\begin{abstract}
    Firstly,a turbofan engine failure is a failure that occurs in multiple components and is of great complexity. Therefore, we used fault tree analysis (FTA) to build a model and determined the importance of ten fault types. Finally, based on the model and importance The cause and consequences of the failure are analyzed in detail and displayed in the form of charts.
    Secondly,in the realm of predictive maintenance, the estimation of Remaining Useful Life (RUL) is a cornerstone for ensuring efficiency and reliability, particularly for instruments with sophisticated structures such as turbofan engine. \textbf{RF with GridSearchCV} is an optimized \textbf{Random Forest} algorithm. Random Forest is  proposed by Breiman in 2001.   and has the advantages of a ability to process large volumes of data and resilience against overfitting. This characteristic is vital in predictive maintenance, where the accuracy of RUL predictions directly impacts operational efficiency and safety. Moreover, RF's intrinsic capability to rank the significance of various features in the dataset provides invaluable insights. This aspect enables maintenance teams to identify and prioritize the factors most critically affecting the lifespan of machinery, leading to more informed decision-making\textsuperscript{\cite{ref3}}.
    Random Forest here operates by constructing multiple \textbf{regression trees} during training and aggregating their predictions. A regression tree is a decision tree variant designed for predicting continuous numerical values. It operates on the same principle as a decision tree but differs in its output, focusing on quantitative outcomes rather than categorical classifications. The structure of the a regression tree starts from a root node that represents the entire dataset, and from this root, the tree branches out based on specific conditions. These conditions are determined in a way that minimizes the variability of the response variable in the resulting subsets, often using mean squared error (MSE) as a key criterion.
    Thirdly,since turbofan engine faults occur on multiple components, we assign different maintenance levels according to the importance of components, namely emergency maintenance, priority maintenance and routine maintenance levels. According to the characteristics of different parts, we take different maintenance methods, that is, post-maintenance, preventive maintenance and predictive maintenance methods. Based on the second question, we have calculated the remaining life of the parts. For the key parts, we adopt the predictive maintenance method, establish the delay time model and maintenance optimization model, and optimize the maintenance cost and maintenance cost. Finally, according to the model and algorithm, the proper maintenance interval is optimized, and the basic maintenance support strategy is formulated.
    Finally,We have given a specific maintenance plan and attached a work guide for maintenance personnel on the last page to help them perform more accurate and efficient maintenance.
    \vspace{5pt}
    \par \textbf{Keywords}:turbofan engines,Fault tree analysis model,regression trees,Simulation optimization,Maintain optimization model

\end{abstract}

\maketitle  % 生成 Summary Sheet
\tableofcontents  % 生成目录

\section{Introduction}
\subsection{Background}
The most complicated system aboard an aircraft is the aero-engine, which powers the aircraft and moves it forward. The engine is commonly referred to as the '' heart of the aircraft '' because it serves as the aircraft's source of thrust as well as supplying energy and electricity for the power supply and hydraulics.
The health of a turbofan engine, a type of jet engine that is frequently used to propel aircraft, has a significant influence on flight safety. 
Information technology is driving the constant increase in complexity of the equipment intelligence, module integration, and system structure of turbofan engines.
This calls for the development and application of more potent, effective, and adaptable maintenance techniques. 
We provide targeted maintenance and guarantee advice along with associated maintenance and guarantee programs using simulation and evaluation tools.

\subsection{Restatement of the Problem}
Taking a certain type of equipment as the research object, here we choose a turbofan engine, find data by ourselves, and analyze its fault characteristics and rules. Use simulation evaluation methods to propose corresponding maintenance support strategies and provide targeted maintenance support suggestions. Specifically, what we should do:
\par Apply fault tree analysis (FTA) and other methods to the research objects (such as typical components such as turbofan engines) to analyze their possible failure modes, failure causes and failure consequences.
\par Combined with the fault data accumulated during use and maintenance of the research object (such as typical components such as turbofan engines), statistical methods and tools are applied to analyze its fault patterns (such as fault distribution, etc.).
\par Based on the failure patterns of research objects (such as typical components such as turbofan engines), maintenance analysis methods such as reliability-centered maintenance (RCMA) and predictive maintenance (PdM) are applied to determine the type of maintenance work and maintenance intervals through simulation optimization and other methods. and maintenance levels, and formulate basic maintenance support strategies.
\par Based on the typical maintenance activities of the research object (such as typical components such as turbofan engines), specific maintenance support suggestions are given: Apply the maintenance work analysis (MTA) method to formulate specific maintenance steps, required personnel and other maintenance support elements; Based on the research objects, suggestions are made for the construction of maintenance resources such as maintenance support personnel and support equipment.

%\section{Assumptions and Justification}

\section{Notations}
The primary notations used in this paper are listed in Table 1. There can be some othernotations to be described in other parts of the paper.
\\
\begin{tabular}{cccccc}
    \toprule
    \makebox[0.2\textwidth][c]{Symbol} & \makebox[0.2\textwidth][c]{Definition} & \makebox[0.2\textwidth][c]{Unit}  \\
    \midrule
    u & Normal phase time & h   \\
    v & Delay Time Indicates the phase time & h  \\
    T & Detection interval of the system & h  \\
    τ & Maximum number of total detection times & time  \\
    TC & Maximum available time & h  \\
    $c_{in}$ & Testing cost &\$  \\
    $c_{p}$ & Preventive maintenance cost &\$  \\
    $c_{r}$ & Preventive replacement  cost &\$  \\
    $c_{c}$ & Recovery cost &\$  \\
    $c_{in}$ & Testing cost &\$  \\
    $c_{p}$ & Preventive maintenance cost &\$  \\
    $c_{r}$ & Preventive replacement  cost &\$  \\
    $c_{c}$ & Recovery cost &\$  \\
    $t_{in}$ & Testing time &h  \\
    $t_{p}$ & Preventive maintenance time &h  \\
    $t_{r}$ & Preventive replacement time &h  \\
    $t_{c}$ & Recovery time &h  \\
    $T_{DOWN}$ & Total down time &h  \\
    \bottomrule
 \end{tabular}

\section{Overall Planning Model}
\subsection{Model Overview}

\subsection{Sub-Model 1:Fault tree analysis model on turbofan engines}
\subsubsection{Overview of sub-model 1}
A turbofan engine failure is a failure that occurs in multiple components and is of great complexity. Therefore, we used fault tree analysis (FTA) to build a model and determined the importance of ten fault types. Finally, based on the model and importance The cause and consequences of the failure are analyzed in detail and displayed in the form of charts.
\subsubsection{Introduction and development of a FTA model for turbofan engines}
Over the years, several methodologies have been developed to facilitate
safety and reliability analysis of systems. Among them, Fault Tree Analysis
(FTA) is one of the oldest and most popular techniques widely used to perform
safety and reliability analysis of systems. In traditional FTA, systems and their
components are usually consider to have two states: working and failed. To
model the logical interaction between different failure events Boolean AND
and OR gates are used, and the causes of system failure are determined in the
form of combinations of events\textsuperscript{\cite{ref1}}.
A fault tree is a logical causality diagram where the elements of the composition are events and logic gates: events are used to describe the state of the system and meta-component failures; logic gates link the events and represent the logical relationships between them.
\par
We have compiled 10 additional common failure modes for turbofan engines through data collection; these failure modes are categorized based on the components of the engines.
The turbofan engine failure is the top event in the fault tree model. The layer 2 intermediate event sub-nodes include airway performance failure, support system failure, rotor failure, and sliding oil system failure; the layer 3 bottom event nodes are  \textit{X}_{1} {\ } {\mathrm{to}} {\ } \textit{X}_{10}.  
\par
Figure 1 displays the fault tree model for the turbofan engine, and Table 1 lists the failure modes denoted by  \textit{X}_{1} {\ } {\mathrm{to}} {\ } \textit{X}_{10}.

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.9\textwidth]{/picture_1/FTA_01.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Fault tree modeling of turbofan engine systems} %最终文档中希望显示的图片标题
    \label{Fig.1} %用于文内引用的标签
\end{figure}


\begin{table}[hp] %%参数： h:放在此处 t:放在顶端 b:放在底端 p:在本页
	\renewcommand\arraystretch{1.2}
	%\setlength{\abovecaptionskip}{0.cm}
	%\setlength{\belowcaptionskip}{-0.cm}
	\centering  % 显示位置为中间
	\textbf{Table 1}~~Items of the fault tree\\  %%表的标题
	\begin{tabular}{p{45pt}cc|cc} %第一列设置宽度为45pt 全为左对齐 没有分割线
		%\setlength{\tabcolsep}{20mm}
		\hline  % 表格的横线
		\toprule % 顶部线
		No. & & Event Name & No. & Event Name \\%[3pt]只改一行    %%表格第一行标题 % 表格中的内容，用&分开，\\表示下一行
		\hline  % 表格的横线
		%\midrule % 中部线
		$\textit{X}_{1}$   & &Component Deformation           & \textit{X}_{6} & Compressor Failure \\    %%表格内容
	    $\textit{X}_{2}$   & &Bearing Failure                 & \textit{X}_{7} & Fan Blade Fault    \\
        $\textit{X}_{3}$   & &Rotor Rub                       & \textit{X}_{8} & Rotor Corrosion and Wear \\
        $\textit{X}_{4}$   & &Combustor Fault                 & \textit{X}_{9} & Turbine Blade Chunking Fault \\
        $\textit{X}_{5}$   & &Turbine Fault                   & \textit{X}_{10} & Oil System Fatigue Wear \\	
		\bottomrule % 底部线
		\hline  % 表格的横线
	\end{tabular}
\end{table}

\par
Determining the minimal set of all cuts in the fault tree model is the primary goal of fault tree analysis. In order to solve the minimum cut-set, the model uses the Fussell approach, the fundamentals of which are described in reference 2 \textsuperscript{\cite{ref2}}. The minimum cut-set of the turbofan engine system fault tree is : \textit{X}_{1} {\ } {\mathrm{to}} {\ } \emph{X}_{10},\textnormal{as the fault tree is primarily composed of "OR" relationships.}

\subsubsection{Modeling Event Importance for FTA}
A turbofan engine is a multi-component, intricate system. A turbofan engine can malfunction due to damage to a single component or as a result of multiple components failing at the same time, which is known as combined failure. As is common knowledge, not every component failure results in the failure of the turbofan engine as a whole. Some components may fail, but not to the point where the engine shuts down or is scrapped. However, the failure of some essential components will cause the engine to fail and possibly shut down. 
We have determined the significance of these ten failure types to the overall failure of turbofan engines by analyzing a great deal of data and consulting relevant literature. Figure 2 shows the classes into which we have divided the importance of these ten defect types.

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.7\textwidth]{picture_1/level_01.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Importance levels for ten fault types} %最终文档中希望显示的图片标题
    \label{Fig.2} %用于文内引用的标签
\end{figure}

\subsubsection{Analyze the cause of the malfunction and the consequences of the malfunction}
Based on the charts, the previous section listed the ten common failure modes of a turbofan engine. In this section, we'll talk about the causes and effects of these ten failure modes.
First, let's examine the reasons behind the failure modes at importance level 1. Event Combustion Chamber Failure: An unstable or insufficient fuel supply may result in incomplete combustion; 
failure may also be brought on by worn-out or damaged combustion chamber components, such as nozzles and spark plugs. The high temperature and pressure inside the combustion chamber may cause overheating and damage to the components.
When the internal parts of the combustion chamber, such as nozzles, spark plugs, etc., are damaged or worn out, it may also lead to failure. The occurrence of a combustion chamber failure can have a very serious adverse effect on a turbofan engine and may result in a reduction of engine thrust, affecting the performance of the vehicle, and improper combustion may result in unstable engine vibration, which can reduce fuel efficiency and even pose a threat to flight safety.
For Event:Turbine Failure, turbine failure is subdivided into high-pressure turbine failure and low-pressure turbine failure, in which the first point of the cause of high-pressure turbine failure is due to the high temperature and high pressure in the working environment of the turbine blades may lead to metal fatigue and thermal expansion, which accelerates the deterioration and damage of the parts, the second point of the lubrication system failure may lead to the increase of the friction of the high-pressure turbine components, which damages the turbine, and the third point of high-temperature environment, the bad materials or corrosion of external substances may damage the surface of the turbine. 
The consequences include: causing engine performance to drop, reducing thrust output, causing imbalance and vibration of the turbine, and causing an abnormal rise in gas temperature. Since it is slightly cumbersome to express in words, in order to give the reader a clearer and more intuitive understanding of the causes and consequences of these ten failure modes, we summarize them in a mind map, which is shown in Figure 3.

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=1.0\textwidth]{picture_1/smartfigure3.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Ten Failure Modes Causes and Consequences.(Note: The left side of each topic center corresponds to the cause, and the right side corresponds to the consequence.)} %最终文档中希望显示的图片标题
    
    \label{Fig.3} %用于文内引用的标签
\end{figure}

\subsection{Sub-Model 2:RF with GridSearchCV}
\subsubsection{Introduction to RF with GridSearchCV}
In the realm of predictive maintenance, the estimation of Remaining Useful Life (RUL) is a cornerstone for ensuring efficiency and reliability, particularly for instruments with sophisticated structures such as turbofan engine. \textbf{RF with GridSearchCV} is an optimized \textbf{Random Forest} algorithm. Random Forest is  proposed by Breiman in 2001.   and has the advantages of a ability to process large volumes of data and resilience against overfitting. This characteristic is vital in predictive maintenance, where the accuracy of RUL predictions directly impacts operational efficiency and safety. Moreover, RF's intrinsic capability to rank the significance of various features in the dataset provides invaluable insights. This aspect enables maintenance teams to identify and prioritize the factors most critically affecting the lifespan of machinery, leading to more informed decision-making\textsuperscript{\cite{ref3}}.
Random Forest here operates by constructing multiple \textbf{regression trees} during training and aggregating their predictions. A regression tree is a decision tree variant designed for predicting continuous numerical values. It operates on the same principle as a decision tree but differs in its output, focusing on quantitative outcomes rather than categorical classifications. The structure of the a regression tree starts from a root node that represents the entire dataset, and from this root, the tree branches out based on specific conditions. These conditions are determined in a way that minimizes the variability of the response variable in the resulting subsets, often using mean squared error (MSE) as a key criterion.

The Random Forest algorithm builds numerous regression trees on various sub-samples of the dataset and then averages their predictions to produce a more accurate and stable result in the context of regression. It enhances the prediction accuracy and robustness compared to using a single decision tree or regression tree.

To improve the accuracy of Random Forest algorithms, reducing risks of turbofan engine failure, parameter optimization is one of feasible methods. By defining a grid of hyperparameter values, GridSearchCV systematically constructs and evaluates the model formed by each parameter combination. Before the search begins, we need to manually list each hyperparameter's possible values, and the combinations of different values of multiple hyperparameters will eventually form a parameter space. The grid search algorithm will take all the parameter combinations in this space, train the model with them, and finally select the combination with the strongest generalization ability as the final hyperparameters of the model\textsuperscript{\cite{ref4}}.
\begin{figure}[htbp]
	\centering
	\begin{minipage}{0.49\linewidth}
		\centering
		\includegraphics[width=0.9\linewidth]{picture_2/flow_regression.png}
		\caption{Regression Tree Flowchart}
		\label{Fig.4}%文中引用该图片代号
	\end{minipage}
	\begin{minipage}{0.49\linewidth}
		\centering
		\includegraphics[width=1\linewidth]{picture_2/forest.png}
		\caption{Random Forest Schematic Diagram}
		\label{Fig.5}%文中引用该图片代号
	\end{minipage}
	%\qquad
\end{figure}

\subsubsection{Data Description and Preprocessing}
The main purpose of this model is to \textbf{predict the remaining useful life} of turbofan engines. In order to test our model, we used the publicly available dataset of the degradation process of key components of the turbofan engine in the Aeronautical Propulsion System Simulation Platform CMAPSS developed by NASA Ames Research Center. The dataset contains analog sensor data generated by different turbofan engines over time and is commonly used to study the estimation of remaining service life.

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.6\textwidth]{picture_2/engine.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Turbofan engine structure diagram} %最终文档中希望显示的图片标题
    \label{Fig.6} %用于文内引用的标签
\end{figure}

\par
The parameters monitored by the 24 sensors in the dataset are shown below.

\begin{longtable}[!htp]{p{60pt}c|c|c} % Adjust the column specifications as needed
    \renewcommand\arraystretch{1.2}
    \caption*{Table 2: 24 turbofan engine sensors} \label{tab:mytable} \\
    \toprule
    \textbf{No.} & & \textbf{Symbol} & \textbf{Meaning} \\
    \midrule
    \endfirsthead
    \captionsetup{justification=centering}
    \caption*{Table 2: 24 turbofan engine sensors (continued)} \\
    \toprule
    \textbf{No.} & & \textbf{Symbol} & \textbf{Meaning} \\
    \midrule
    \endhead
    
    $1$   & &H            & Flight altitude \\    
    $2$   & &Ma           & Mach number \\ 
    $3$   & &TRA          & Throttle lever angle \\
    $4$   & &T2           & Fan inlet temperature \\
    $5$   & &T24          & Low pressure compressor outlet temperature \\
    $6$   & &T30          & High pressure compressor outlet temperature \\
    $7$   & &T50          & Low pressure turbine outlet temperature \\
    $8$   & &P2           & Fan inlet pressure \\
    $9$   & &P15          & Total pressure of external duct \\
    $10$  & &P30          & High pressure compressor outlet total pressure \\
    $11$  & &NF           & Unmodified fan speed \\
    $12$  & &NC           & Unmodified core machine speed \\
    $13$  & &EPR          & Engine pressure ratio \\
    $14$  & &PS30         & High pressure compressor outlet static pressure \\
    $15$  & &PHI          & Fuel flow and P30 ratio \\
    $16$  & &NRF          & Fan correction speed \\ 
    $17$  & &NRC          & Modified speed of core machine \\
    $18$  & &BPR          & Bypass ratio \\
    $19$  & &FARB         & Combustion chamber gas ratio \\
    $20$  & &HT\_BLEEd    & Bleed air enthalpy \\
    $21$  & &NF\_DMD      & Fan speed command value \\
    $22$  & &PCNFR\_DMD   & Fan speed correction command value \\
    $23$  & &W31          & High pressure turbine cooling air flow \\
    $24$  & &W32          & Low pressure turbine cooling air flow \\
    \bottomrule
\end{longtable}

\par
To see the lifetime distribution of the 100 engines in the dataset, we first create a kernel density distribution map. 

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.6\textwidth]{picture_2/kde.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Kernel Density Estimation of Full Useful Life } %最终文档中希望显示的图片标题
    \label{Fig.7} %用于文内引用的标签
\end{figure}
\par
After that, we worked on feature engineering to further refine the dataset with the goal of creating or extracting characteristics from the unprocessed data that are essential to improving the performance of the model. This required an elimination of sensors that showed no relationship to changes in Remaining Useful Life (RUL) for the CMAPSS dataset. From the box plots, it can be seen that some of the sensors' values change independently of the engine degradation state. Furthermore, we removed sensors that showed high redundancy because of their robust correlation,  so that our predictive modeling efforts would utilize a more efficient and simplified dataset.

%子图
\begin{figure}[htbp]
	\centering
	\begin{minipage}{0.4\linewidth}
		\centering
		\includegraphics[width=0.9\linewidth]{picture_2/box.png}
		\caption{Boxplot of Sensor Variation over Lifecycle}
		\label{Fig.8}%文中引用该图片代号
	\end{minipage}
	\begin{minipage}{0.4\linewidth}
		\centering
		\includegraphics[width=1\linewidth]{picture_2/cor.png}
		\caption{Heatmap of Sensor Correlations}
		\label{Fig.9}%文中引用该图片代号
	\end{minipage}
	%\qquad
\end{figure}

\par
Previous research has demonstrated that the segmented linear deterioration model is more effective at managing the RUL prediction issue for the CMAPSS dataset.  The segmented linear degeneracy model is also used in our preprocessing of the dataset to establish the RUL labeling, which is set to 125 flight cycles.\textsuperscript{\cite{ref5}}
\par
Following the preprocessing of the data, Fig. illustrates the link between the sensors and the remaining engine life. The values that are tracked by every sensor at the conclusion of its life cycle are aberrant to varied degrees. 

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.5\textwidth]{picture_2/time2.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Engine Life Cycle} %最终文档中希望显示的图片标题
    \label{Fig.10} %用于文内引用的标签
\end{figure}
\par
Based on the time series plots and kernel density distribution plots, we performed a K-S test for the lifetime distribution of the pairs of turbine engines against several common survival functions, and according to the results, the best fit was found to be with the lognormal distribution. This distribution is suitable for describing random variables whose natural logarithms are normally distributed, especially those distributions with long tails that are often encountered in real data. The lognormal distribution distribution has the following distribution function and probability density function.
\\PDF:
\\$$f(x; \mu, \sigma) = \frac{1}{x \sigma \sqrt{2\pi}} e^{-\frac{(\ln(x) - \mu)^2}{2\sigma^2}}$$
\\CDF:
\\$$F(x; \mu, \sigma) = \frac{1}{2} \left[1 + \text{erf}\left(\frac{\ln(x) - \mu}{\sigma \sqrt{2}}\right)\right]$$
\\error function:
\\$$\text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} \, dt$$

\begin{table}[h]
    \caption*{Table 3: Kolmogorov-Smirnov Test Results}
    \centering
    \label{tab:ks-test}
    \begin{tabular}{|c|c|c|}
    \hline
    & Statistic & P-Value \\
    \hline
    Log-Normal & 0.094024 & 0.319407 \\
    \hline
    Weibull & 0.156366 & 0.013329 \\
    \hline
    Exponential & 0.469347 & 1.007432e-20 \\
    \hline
    \end{tabular}
\end{table}
\par
Maximum likelihood estimate can be used to approximate the form and scale parameters. The parameters fitted to the available dataset are as follows. Once the parameters were determined, we drew the QQ graph to show the fitting outcomes. 
\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=1\textwidth]{picture_2/lognorm.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Lognormal Distribution Function} %最终文档中希望显示的图片标题
    \label{Fig.11} %用于文内引用的标签
\end{figure}
\par
\begin{table}[h]
    \caption{Estimate of parameters}
    \centering
    \label{Estimate of parameters}
    \begin{tabular}{|c|c|c|}
    \hline
     Shape Parameter & \sigma & 0.213151 \\
     \hline
     Scale Parameter & e^\mu  & 200.569583 \\
     \hline
    \end{tabular}
\end{table}
\par

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.6\textwidth]{picture_2/qq.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Log-Normal QQ Plot} %最终文档中希望显示的图片标题
    \label{Fig.12} %用于文内引用的标签
\end{figure}
\par

\subsubsection{Model Results and Evaluation}
We utilized the above data and input the feature sequences into the Random Forest model for training, where the training and test sets are divided using the original dataset. The random forest is able to automatically identify and assign different weights to different feature sensors by integrating multiple regression trees and performing extensive correlation calculations using historical data. Ultimately, the model outputs predicted values for the remaining life of the turbofan engine calculated based on these weights. Subsequently, we used grid search optimization to train the training set again with a view to obtaining the best parameter.
For the purpose of verifying the reliability of the Random Forest with GridSearchCV model, we used RMSE and $R^2$ of the prediction results as the measurement indicators of the model accuracy. Before and after the parameter optimization, and for the datasets with two different failure scenarios, 1 and 3, the results of the calculation of each metric in the training and validation sets are as follows:

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.6\textwidth]{picture_2/perfprmance.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Model Performance} %最终文档中希望显示的图片标题
    \label{Fig.13} %用于文内引用的标签
\end{figure}
\par

Since the RMSE is smaller and $R^2$ is larger, we can judge the reliability of the Random Forest with GridSearchCV model's predictions. The calibration curves of the observed and projected values also support this result.

%子图
\begin{figure}[htbp]
	\centering
	\begin{minipage}{0.49\linewidth}
		\centering
		\includegraphics[width=1\linewidth]{picture_2/DF.png}
	\end{minipage}
    \label{Fig.14}
	\begin{minipage}{0.49\linewidth}
		\centering
		\includegraphics[width=1\linewidth]{picture_2/DFT.png}
	\end{minipage}
	\caption{Calibration Curves of Model Before and After Optimization}
        \label{Fig.14}
\end{figure}

Because of the model's dependability, we can list each feature's weight to gauge its significance, which helps with maintenance decisions for turbofan engines.

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.5\textwidth]{picture_2/shap.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Feature Contribution} %最终文档中希望显示的图片标题
    \label{Fig.16} %用于文内引用的标签
\end{figure}
\par

In our study, we successfully applied a random forest model combined with GridSearchCV optimization to predict the RUL of turbofan engines. The Random Forest algorithm's integrated learning qualities, along with GridSearchCV's system parameter optimization, enabled the model to demonstrate efficient and accurate prediction skills on NASA's CMAPSS dataset. The successful application of this model not only provides dependable support for turbofan engine health monitoring and predictive maintenance, but also establishes an effective methodological baseline for life prediction tasks of similar complex systems.


\subsection{Sub-Model 3:}
\subsubsection{Overview of sub-model 3}
Since turbofan engine faults occur on multiple components, we assign different maintenance levels according to the importance of components, namely emergency maintenance, priority maintenance and routine maintenance levels. According to the characteristics of different parts, we take different maintenance methods, that is, post-maintenance, preventive maintenance and predictive maintenance methods. Based on the second question, we have calculated the remaining life of the parts. For the key parts, we adopt the predictive maintenance method, establish the delay time model and maintenance optimization model, and optimize the maintenance cost and maintenance cost. Finally, according to the model and algorithm, the proper maintenance interval is optimized, and the basic maintenance support strategy is formulated.\textsuperscript{\cite{ref6}}
\subsubsection{Introduction and development of delay time model}
\subsubsubsection{Problem analysis and maintenance strategy}
\ 
\newline
\indent In the condition maintenance model, the 
stochastic process method can directly 
describe the degradation process
. In the 
actual maintenance work, it is often 
difficult to define the specific degradation 
state, so the degradation state needs to be
dissociated. delay time model 
is widely used in multi-stage degradation, 
which was first proposed and applied to the 
production line maintenance problem in 1984.
At present, the model is widely used in 
overhaul planning optimization.For this problem, we consider the reliability model of incomplete detection and incomplete maintenance to effectively describe the degradation process and the probability of failure. With the goal of minimizing the system cost rate, the maintenance plan optimization model is constructed, and the system maintenance and replacement strategies are optimized by analyzing the detection results and failure probability, considering the practical constraints such as reliability and availability. Finally, the necessary conditions for the application of the maintenance strategy are obtained by comparing with the maintenance strategy that does not consider detection in the existing research, and the maintenance cycle is determined.
Based on the first question, we determined the importance levels of ten defect types. On the basis of the importance, we also considered the factors of maintenance cost, and determined the maintenance types of these ten defects respectively, as shown in the figure below:
\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.8\textwidth]{picture_3/wxfl.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Classification of turbofan engine system fault maintenance types} %最终文档中希望显示的图片标题
    \label{Fig.17} %用于文内引用的标签
\end{figure}
For turbofan engine component failure, we adopt three maintenance methods: preventive maintenance, scheduled maintenance and fault repair maintenance. Preventive maintenance can prevent failures by regularly checking and replacing parts; Scheduled maintenance can be carried out according to the equipment use time or operation cycle; The fault repair maintenance is the direct repair after the fault occurs. Since the combustion chamber, turbine and compressor are key components in the performance of the gas path, once failure occurs, it will lead to shutdown, resulting in huge losses. Meanwhile, turbine blades are expensive and the overall replacement cost is large after failure, so we adopt predictive maintenance for these four components. Fan blades and bearings are relatively inexpensive, and the use of sensors to monitor their condition in real time and calculate their remaining life is more expensive than regular direct replacement, so preventive maintenance is used for these two components; Parts deformation failure, machine turn rub failure, rotor corrosion and wear failure and lubricating oil system fatigue wear failure will not cause the turbine engine shutdown impact, while the price is relatively low, the use of the system life of direct replacement is the most economic choice, so for these four failures we choose fault repair maintenance method.
\par
The maintenance cycle of the preventive maintenance method is determined by experience. Now we discuss the optimization of the maintenance cycle of the predictive maintenance method. According to the delay time model , set 
the normal phase time as u, the delay time 
phase time as v, and the system life at this 
time is u+v. The density function and 
cumulative distribution of defined random 
variables u and v are $f_{1}(u), F_{1} (u)$ and  $f_{2}(u), F_{2} (u)$,respectively, and the corresponding 
incidence of the two stages are λ (u) and $\eta(u)$, 
which is called the defect rate and the 
failure rate after the occurrence of defects.
The maintenance plan of the system depends on 
the status of the inspection, when the 
inspection requires cost, it is necessary to 
determine the optimal one first inspection strategy. In addition, incomplete 
maintenance leads to system replacement 
after multiple repairs, so another 
important part of the overhaul plan is the 
replacement strategy. 
\par
Let the system be inspected with a fixed 
period T, and the detection can only 
identify the defects that have occurred with 
probability r (0<r≤1), but will not identify 
the normal situation as a defect. When the 
defect is detected, the system carries out 
preventive maintenance, and the defect will 
be repaired after maintenance. If no defect 
is detected, no repair is performed. The 
maintenance is incomplete, so the system can 
not be restored to the new state after 
maintenance. The effective age method is 
used to describe the state after 
maintenance, and the age regression factor 
is a (0≤a≤ 1), indicating the reduction 
proportion of the effective age of the 
system after maintenance. If the cumulative 
running time of the system reaches the 
maximum available time TC or the cumulative 
number of tests reaches the upper limit of 
the number τ, then the pre-ventive replacement
is carried out, and the state of the system 
after replacement is new. The maximum 
available time is the inherent attribute of 
the system, also known as the technical life. 
If the cost can be reduced by replacement at 
a previous time, then this time is called 
the economic life
. When the system fails 
in operation, it is restored to operational 
condition by minimal repair, which does not 
change the degradation of the system, and 
the defect must be eliminated by preventive 
maintenance at the next inspection. 
\par
In the above process, the costs of 
testing, preventive maintenance, preventive 
replacement, and minimum maintenance (i.e., 
restorative maintenance) are denoted as $c_{in} , c_{p} ,c_{r},c_{c}$,and the corresponding downtime is 
denoted as$t_{in}, t_{p},t_{r},t_{c}$,The minimum 
maintenance cost includes the expected loss 
cost caused by the failure, so it is higher 
than the cost of preventive maintenance and 
replacement. In the case of known TC, the 
maintenance strategy is defined as the 
detection period T and the detection upper 
limit number τ. When the detection period is 
too long, the defects can not be effectively 
identified, and the system is prone to 
failure; If the cycle is too short, 
additional maintenance costs will be added.
It is necessary to minimize the average 
system maintenance cost by determining 
reasonable inspection intervals and 
replacement strategies.
\par
In order to facilitate model 
construction, the following 
hypotheses are proposed.
\par
Hypothesis 1 \hspace{1cm}The initial 
state of the system is brand 
new
\par
Hypothesis 2 \hspace{1cm} The system has the minimum 
reliability constraint in an update cycle, 
which represents the maximum failure 
probability constraint in practice.
\par
Hypothesis 3 \hspace{1cm}The system downtime caused by 
maintenance or failure generates downtime 
loss costs. In order to ensure the usable 
time of the system, the system has a 
minimum availability constraint.
\par
Hypothesis 4 \hspace{1cm}The system 
maintenance time and troubleshooting time 
are short, so the effect of downtime on 
system degradation is not considered.
\subsubsubsection{Maintenance optimization model}
\ 
\newline
\indent Maintenance optimization models 
typically include a reliability model (i.e., 
a degradation model) and a maintenance 
schedule optimization model
. The first 
construction consideration is not complete
Reliability model for full overhaul. The 
incompleteness of detection makes 
preventive maintenance a random event, and 
the degradation rate of the system is 
different after maintenance at different 
time. In addition, incomplete detection may 
prolong the existing defects to the next 
inspection interval, and the system state 
is not enough Markov, so it is necessary to 
build a reliability model based on 
recurrence relationship.\\
\\
\textbf{2.1 Reliability model}
\newline
2.1.1 Service age rollback model \hspace{0.3cm} Describes the 
effect of incomplete maintenance through 
the service age rollback factor. If the 
cumulative life of the system aT the KTH 
maintenance (time is $t_k$) is $T_k$, then the 
accumulated life of the system after 
incomplete maintenance (that is, the 
effective service age) is restored to $aT_k$.
Assuming that the draft age rollback factor 
acts on both the normal stage and the delay 
time stage, the defect rate $\lambda_{k+1}(u) $ and the 
independent fault rate $ h_{k+1}(u)$ of the system 
after maintenance at $t_k$ are respectively 
changed as follows.
\begin{equation}
    \begin{aligned}
    \lambda_{k+1}(u) =
    \begin{cases}
        \lambda_{k}(u+aT_{k})=\lambda(u+aT_{k}) & \text{Maintenance at } t_k \text{ time} \\
        \lambda_{k}(u+T_{k}) & \text{No maintenance.}
    \end{cases}
    \end{aligned}
    \end{equation}
\begin{equation}
        \begin{aligned}
        \eta_{k+1}(v) =
        \begin{cases}
            \eta_{k}(v+aT_{k})=\eta(v+aT_{k}) & \text{Maintenance at } t_k \text{ time} \\
            \eta_{k}(v+T_{k}) & \text{No maintenance.}
        \end{cases}
        \end{aligned}
        \end{equation}
        Where: $T_{k}$ is the cumulative use time of 
the system since the last repair\\
2.1.2 The probability of defect detection \hspace{0.3cm}
  Consider when the system is detected at t (i>0) and 
the probability of defect detection and 
preventive maintenance is  $P_{d}(t_i)$. Let the time of last 
preventive maintenance be t (1<k≤i−1), as shown in 
Figure 18.
$g_{k+1}(u)$ is the probability density function corresponding to the failure not detected before $t_{i}-t_k-u$ time after k repairs.
The probability that a defect $(t_{k} \leq t_{l-1} < t_{l} \leq t_{i})$ 
occurred at some time $(t_{l-1}, t_{l})$ and that no fault occurred 
before the detection time $t_{i}$ is as follows.
\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.7\textwidth]{picture_3/Sketch map of defect detection.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Sketch map of defect detection} %最终文档中希望显示的图片标题
    \label{Fig.5} %用于文内引用的标签
\end{figure}
\begin{equation}
    {P_{\rm{d}}}({t_i}|{t_k}) = r\sum\limits_{l = k + 1}^i {\left\{ {{{(1 - r)}^{i - l}}\int_{{t_{l - 1}} - {t_k}}^{{t_l} - {t_k}} {{g_{k + 1}}(u)[1 - {F_{k + 1}}({t_i} - {t_k} - u)]{\rm{d}}u} } \right\}}
    \end{equation}
If the detection probability 
r=1, then equation (3) is 
rewritten as
\begin{equation}
    {P_{\rm{d}}}({t_i}|{t_k}) = \int_{{t_{i - 1}} - {t_k}}^{{t_i} - {t_k}} {{g_{k + 1}}(u)[1 - {F_{k + 1}}({t_i} - {t_k} - u)]{\rm{d}}u} 
\end{equation}
That is, the detected defect can only occur 
in the time period $(t_{i−1} , t_i)$.

2.1.3 Failure probability  \hspace{0.3cm} Consider that the last preventive maintenance occurred at $t_k$ time, the failure situation at the $t_d$ time of the system in the detection interval $(t_{i−1}, t_i)$
, as shown in Figure 
6. The defect may occur at some point in $(t_{l-1},t_l)$ where  $(t_{k} \leq t_{l-1} < t_{l} \leq t_{i})$ . The probability of this occurrence is as follows
\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.7\textwidth]{picture_3/Sketch map of failure occurrence.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Sketch map of failure occurrence} %最终文档中希望显示的图片标题
    \label{Fig.6} %用于文内引用的标签
\end{figure}
\begin{equation}
    \begin{array}{l} {P_{\rm{f}}}({t_i}|{t_k}) = \sum\limits_{l = k + 1}^i {\left\{ {{{(1 - r)}^{i - l}}\int_{{t_{l - 1}} - {t_k}}^{{t_l} - {t_k}} {{g_{k + 1}}(u)} \times } \right.} \\ \left. {[{F_{k + 1}}({t_i} - {t_k} - u) - {F_{k + 1}}({t_{i - 1}} - {t_k} - u)]{\rm{d}}u} \right\}. \end{array}
\end{equation}
After the system fails in the $(t_{i −1} , t_i) $
period, the defect is eliminated by preventive 
maintenance at the $t_{i}$ hour. The conditionali 
probability of preventive maintenance at time $t_{i}$  is
\begin{equation}
    {P_{\rm{f}}}({t_i}|{t_k}) = \sum\limits_{l = k + 1}^i {\left\{ {{{(1 - r)}^{i - l}}\int_{{t_{l - 1}} - {t_k}}^{{t_l} - {t_k}} {{g_{k + 1}}(u) [ {F_{k + 1}}({t_i} - {t_k} - u) - {F_{k + 1}}({t_{i - 1}} - {t_k} - u)]{\rm{d}}u} } \right\}}
    \end{equation}
The probability of preventive 
maintenance at time  $t_{i}$ is
\begin{equation}
    \begin{split} {P_{\rm{m}}}({t_i}) = & \sum\limits_{k = 0}^{i - 1} {{P_{\rm{m}}}({t_k}){P_{\rm{m}}}({t_i}|{t_k})} = \sum\limits_{k = 0}^{i - 1} {{P_{\rm{m}}}({t_k})[{P_{\rm{d}}}({t_i}|{t_k}) + {P_{\rm{f}}}({t_i}|{t_k})]} . \end{split}
\end{equation}
At the beginning of system use (k=0), 
there is $P_m (t_0) =1$.

2.1.4 Reliability function Reliability  \hspace{0.3cm} Reliability function reliability R (t)
represents the probability that the system 
will not fail from the initial time to the time t. To calculate the reliability at any 
time, consider the probability that no failure 
will occur at time $t_{i-1}$ to time t $(t_{i-1}<t <t_i)$.
In the case that the last repair occurred at $t_k$ time, the probability of the system failing at the time period $(t_{i−1}, t)$ is as follows:
\begin{equation}
    {P_{\rm{f}}}(t|{t_{i{\rm{ - }}1}}) = \sum\limits_{k = 1}^{i - 1} {{P_{\rm{m}}}({t_k})} {P_{\rm{f}}}(t|{t_k};{t_{i{\rm{ - }}1}}).
\end{equation}
Where:
\begin{equation}
    \begin{split} {P_{\rm{f}}}(t|{t_k};{t_{i - 1}}) = & \int_0^{t - {t_{i - 1}}} {{g_{k + 1}}(u){F_{k + 1}}(t - {t_{i - 1}} - u){\rm{d}}u} + \\ & \sum\limits_{l = k + 1}^{i - 1} {\left\{ {{{(1 - r)}^{i - l}}\int_{{t_{l - 1}} - {t_k}}^{{t_l} - {t_k}} {{g_{k + 1}}(u) \times } } \right.} \\ & \left. {[{F_{k + 1}}(t - {t_k} - u) - {F_{k + 1}}({t_{i - 1}} - {t_k} - u)]{\rm{d}}u} \right\}. \end{split}
\end{equation}
$(t_{i−1} , t) $the probability of no 
failure is:

\begin{equation}
    {P_{\rm{R}}}(t|{t_{i - 1}}) = 1 - {P_{\rm{f}}}(t|{t_{i - 1}}).
\end{equation}
Therefore, the reliability of the 
system at time $t(t>t_{i-1})$ is
\begin{equation}
    R(t) = R({t_{i - 1}}){P_{\rm{R}}}(t|{t_{i - 1}}).
\end{equation}

\textbf{2.2 Parameter estimation method}
\ 
\newline
\indent In the incomplete overhaul model, in addition to the defect distribution and fault distribution parameters, the detection probability and service age regression factor should also be determined. Based on literature , a parameter estimation method is proposed when the maintenance and detection intervals are not fixed.
\\
\indent In actual inspections, there are two situations: maintenance and no maintenance. Let the actual maintenance time sequence of the system be $(t_1^0,t_2^0, \cdots ,t_n^0)$, and assume that there are m(k) undetected defects between adjacent maintenance times $t_k^0{\text{, }}t_{k + 1}^0$, with the next detection time of $t_{k,m(k)}^0$ (k<n) being $t_{k + 1}^0$. The probability of detecting a defect at inspection time $t_{k,i}^0$, where $i = 1,\cdots,m(k)$ and $k = 1,\cdots,n - 1$, is given by:
\begin{equation}
    \begin{split} & {P_{\rm{d}}}(t_{k,i}^0|t_k^0) = r\sum\limits_{l = 1}^i {\left\{ {{{(1 - r)}^{i - l}}\int_{t_{k,l - 1}^0 - t_k^0}^{t_{k,l}^0 - t_k^0} {{g_{k + 1}}(u) \times } } \right.} \\ & {[1 - {F_{k + 1}}(t_{k,i}^0 - t_k^0 - u)]{\rm{d}}u} \Bigg\};\;i = 1,\cdots,m(k). \end{split}
\end{equation}
The likelihood function 
for the detection 
probability is
\begin{equation}
    {L_1} = \sum\limits_{k = 1}^{n - 1} {\left[\sum\limits_{i = 1}^{m(k)} {\ln\; (1 - {P_{\rm{d}}}(t_{k,i}^0|t_k^0))} + \ln\; ({P_{\rm{d}}}(t_{k + 1}^0|t_k^0))\right]}. 
\end{equation}
The actual system failure sequence is denoted as $(x_1^0,x_2^0, \cdots ,x_m^0)$. Since the occurrence of failures is relatively rare in practice, for the purpose of calculation, the time of failure occurrence is transformed into whether a failure occurs within the inspection interval. That is, $x_{k,i}^0$=1 indicates that a failure occurs between the two consecutive inspections $t_{k,i}^0{\text{, }}t_{k,i + 1}^0$. The probability of this event occurring is given by:
\begin{equation}
    \begin{split} & {P_{\rm{f}}}(x_{k,i}^0|t_k^0) =  \sum\limits_{l = 1}^i {\left\{ {{{(1 - r)}^{i - l}}\int_{t_{k,l - 1}^0 - t_k^0}^{t_{k,l}^0 - t_k^0} {{g_{k + 1}}(u)[{F_{k + 1}}(t_{k,i + 1}^0 - t_k^0 - u) - } } \right.} \\ & \left. {{F_{k + 1}}(t_{k,i}^0 - t_k^0 - u)]{\rm{d}}u} \right\};\; i = 0,\cdots ,m(k). \end{split}
\end{equation}
The likelihood function 
for failure occurrence 
is
\begin{equation}
    \begin{split} {L_2} = & \sum\limits_{k = 1}^{n - 1} {\left\{ {\left( {\sum\limits_{i = 0}^{m(k)} {\ln\; (1 - x_{k,i}^0)} (1 - {P_{\rm{f}}}(x_{k,i}^0|t_k^0))} \right) + } \right.}  \left. {x_{k,i}^0\ln\; ({P_{\rm{f}}}(x_{k,i}^0|t_k^0))} \right\}. \end{split}
\end{equation}
Synthesizing the above, 
the likelihood function 
is constructed:
\begin{equation}
    L = {L_1} + {L_2}.
\end{equation}
\textbf{2.3 Overhaul plan 
optimization model}
\ 
\newline
\indent 2.3.1  Overhaul strategy model \hspace{0.3cm} Under the maintenance strategy, the 
detection interval of the system is T, the 
detection upper limit is τ, and the time of
the system's i detection is
\begin{equation}
    {t_i} = i T;\quad i = 1,2,\cdots,\tau .
\end{equation}
The update period of the system is 
constrained by τ and TC. The indicative 
variable σ is defined, and σ=1 indicates that 
the update period of the system is TC.
\begin{equation}
    \sigma = \left\{ {\begin{array}{*{20}{c}} 1,&{T \tau \geqslant {\rm {TC}},\; R({\rm {TC}}) \geqslant {R_{{\rm{min}}}}}; \\ 0,&{\text{otherwise}}. \end{array}} }\right.
\end{equation}
In the update cycle, the total downtime $T_{DOWN} $
of the system includes the inspection time, 
the preventive maintenance time, the minimum 
maintenance time and the replacement time .
The system conducts a total of τ−1 checks, 
since the replacement will be performed 
directly at the τth check or TC.

\begin{equation}
    \begin{split} T_{\rm{Down}} = & (\tau - 1){t_{\rm{in}}} + \sum\nolimits_{i = 1}^{\tau - 1} {{P_{\rm{d}}}({t_i}) {t_{\rm{p}}} - } \\& {\left[ {\sigma \ln R({\rm{TC}}) + (1 - \sigma )\ln R({t_\tau })} \right] {t_{\rm{p}}} + {t_{\rm{r}}}.} \end{split}
\end{equation}

Availability A is the ratio of the 
run-time to the update cycle:
\begin{equation}
    A(T,\tau ) = 1 - \frac{{{T_{{\rm{Down}}}}(T,\tau )}}{{\sigma \times {\rm{TC}} + (1 - \sigma ) T \tau }}.
\end{equation}
Based on the above analysis and renewal 
income theory, the maintenance plan 
optimization model  is established, 
which takes the minimum average cost rate of 
the system as the goal and considers the 
constraints of reliability and availability. 
The objective function and constraint 
conditions are as follows.
\begin{equation}
    \begin{split} \min\; \bar C(T,\tau ) = & E({C_{{\rm{sum}}}})/E(L) = \\ & \left\{ {(\tau - 1){c_{\rm{in}}} + \sum\limits_{i = 1}^{\tau - 1} {{P_{\rm{d}}}({t_i}) {c_{\rm{p}}}} - [\sigma \ln R({\rm{TC}}) + } \right.\\ & {(1 - \sigma )\ln R({t_\tau })] {c_{\rm{c}}} + {c_{\rm{r}}} + {T_{{\rm{Down}}}}{c_{\rm{d}}}} \Bigg\}\Bigg/\\ & [\sigma \times {\rm{TC}} + (1 - \sigma )T\tau] . \end{split}
\end{equation}
\begin{equation}
    \begin{split} & {\rm{s.t.}}\\& \;\;\;{R_{{\rm{min}}}} \leqslant \left\{ {\begin{array}{*{20}{c}} {R({t_\tau })},&{{\rm{ }}T \tau < \rm TC}; \\ {R(\rm TC)},&{{\rm{ }}T \tau \geqslant \rm TC}; \end{array}} \right. \end{split}
\end{equation}
\begin{equation}
    A(T,\tau ) \geqslant {A_{\min }};
\end{equation}
\begin{equation}
    1 \leqslant T \leqslant {T_{\max }}
\end{equation}
\begin{equation}
    1 \leqslant \tau \leqslant \left\lceil {{\rm{TC}}/T} \right\rceil ;
\end{equation}
\begin{equation}
    T,\tau \in \bf N.
\end{equation}
Equation (21) is the objective function, representing the ratio of the total expected cost \(C_{\text{sum}}\) to the inspection interval \(L\). The cost includes inspection cost, preventive maintenance cost, minimal repair cost, replacement cost, and downtime cost.

Equation (22) represents the minimum reliability constraint, ensuring that the system's reliability at any given time does not fall below \(R_{\text{min}}\).

Equation (23) represents the minimum availability constraint.

Equations (24) to (26) are the decision variable constraints, where the inspection interval and the maximum number of inspections are both positive integers subject to an upper limit constraint. Here, \(T_{\text{max}}\) denotes the time until the system degrades to the minimum reliability without maintenance.\\
\indent Determination of defect identification probability r
\hspace{0.3cm} Through periodic inspection (marking the inspection period as $T_1$), the parts are repaired when the remaining life of the parts is lower than the threshold value, which we mark as $\rho$. Through simple simulation calculation, 
there is a certain inverse relationship between the defect identification probability r and $T_1$. Based on the calculation of pdf in the second question, we can find that the remaining life of the parts is sharply reduced. According to the change trend of pdf of each part,
 we determined the threshold $\rho$ of each part, as shown in the figure below(Figure.20), and calculated the maintenance cycle through the single-objective optimization of reducing the inspection cost and improving the defect identification probability r.
 \begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.7\textwidth]{picture_3/Schematic diagram of thresholds for performing repairs.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Schematic diagram of thresholds for performing repairs} %最终文档中希望显示的图片标题
    \label{Fig.20} %用于文内引用的标签
\end{figure}



\subsection{Sub-Model 4:}
\subsubsection{Turbofan engine simulation optimizes predictive maintenance process}
In the above, we have established three fault analysis models of turbofan engines, given a specific simulation predictive maintenance model, and verified its reliability.
However, we know that in daily maintenance operations, for critical maintenance tasks, it often happens that the preparation work is not careful, resulting in a lack of spare parts and other support during the operation, and the maintenance task has to be turned into a recovery task; sometimes due to safety and other aspects of failure The assessment is comprehensive, resulting in safety injuries to personnel performing tasks; there are also cases where key maintenance tasks are terminated due to lack of cooperating personnel, insufficient preparation for setting up software, etc.; there are also cases where maintenance work verification work is not carried out or is insufficient, resulting in shutdowns event. In the above, we have established three fault analysis models of turbofan engines, given a specific simulation predictive maintenance model, and verified its reliability. Therefore, we introduce MTA (Maintenance task analysis and action) maintenance task analysis and implementation process here to help us complete the entire critical maintenance task.
MTA provides us with a rigorous analysis and implementation process control process for repair and maintenance operations. The MTA implementation flow chart is shown in the figure below. Each step of the simulation optimization fault repair of the turbofan engine is explained below.

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.5\textwidth]{picture_4/MTA.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Leveraging MTA's Turbofan Engine Simulation to Optimize Predictive Maintenance Processes} %最终文档中希望显示的图片标题
    \label{Fig.21} %用于文内引用的标签
\end{figure}

\par 
The final specific maintenance steps, required personnel and other maintenance support elements are implemented as follows:
\par \textbf{Step 1  predictive maintenance:Fixed period inspection and irregular period maintenance method}
\par For the first-stage turbofan engine components, the defect rate and independent failure rate after repair, the defect detection rate and the probability of failure are calculated by establishing a reliability model, and the threshold is determined through the parameter estimation method and the detection interval is not fixed. The fixed detection probability and service age regression factor; the available time is obtained through availability analysis. Consider the reliability indicators of the system obtained by reliability analysis, namely the defect rate and independent failure rate after repair, defect detection rate and probability of failure, etc.; availability analysis obtains the available time; the importance of faulty components and the maintenance of different components Constrained by comprehensive factors such as method and maintenance level, we established a maintenance plan model with the smallest average cost rate, and then determined the maintenance interval through enumeration optimization. Predictive maintenance at this time can achieve a cost-optimized but risk-minimized effect. Professional maintenance personnel need to be assigned regularly to perform predictive maintenance.
\par \textbf{Step 2  preventive maintenance: Fixed period inspection and fixed period maintenance methods}
\par For the turbofan engine components with Level 2 importance, we have determined its maintenance cycle and inspection cycle, which will help workers carry out regular inspections\par\textbf{Step 3  fault repair maintenance: Fixed period inspection, repair only when parts fail} 
\par The occurrence of defects in third-level parts will not have a very serious impact on the turbofan engine (such as directly causing shutdown, etc.), so we adopt the method of repairing if it is broken, and the inspection cycle can be appropriately lengthened, without Testing these parts too often. This results in a less expensive repair method. Personnel assignments do not need to be specific.


\subsubsection{Make suggestions for the construction of maintenance resources}
\textbf{1.Recommendations for maintenance and support personnel:}
\begin{itemize}
    \item It is recommended to establish an experienced maintenance team, including mechanics, electronic technicians and professional engine maintenance engineers.}
    \item Regular training and update courses are provided to ensure that maintenance personnel are aware of the latest maintenance methods and technologies.
\end{itemize}
\par
\textbf{2.Recommendations for maintenance and support equipment:}
\begin{itemize}
    \item Ensure high-quality repair equipment and tools are available to support turbofan engine repair work.
    \item Appropriate equipment including measuring instruments, testing equipment, disassembly and assembly tools and special tools are necessary.
    \item Regularly maintain and calibrate repair equipment to ensure proper functioning and accuracy.
\end{itemize}
 \par
\textbf{3. Suggestions for building maintenance resources:}
\begin{itemize} 
    \item    It is recommended to establish a comprehensive maintenance spare parts inventory to ensure that the required parts and materials can be obtained in a timely manner.
    \item Ensure good relationships with suppliers to obtain high quality spare parts and support.
    \item Implement regular maintenance resource assessments to ensure resource adequacy and effectiveness.
\end{itemize}
\par
Finally, we took the turbofan engine compressor failure as an example and created a maintenance form to arrange the work of maintenance personnel and allocate resources, so as to make our suggestions for the construction of maintenance resources such as support equipment.

\begin{figure}[H] %H为当前位置，!htb为忽略美学标准，htbp为浮动图形
    \centering %图片居中
    \includegraphics[width=0.9\textwidth]{picture_4/guide.png} %插入图片，[]中设置图片大小，{}中是图片文件名
    \caption{Turbofan Engine Compressor Maintenance Work Guide} %最终文档中希望显示的图片标题
    \label{Fig.22} %用于文内引用的标签
\end{figure}


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